I'm new to the forum, so I apologize if I am in the wrong section!
I'm studying Claes Johnson's book "Numerical solution of partial differential equations by the Finite Element Method", and I'm trying to solve the proposed problems.
What is not very encouraging is that I'm already stuck at problem 1.1. :-) It seems easy and I understand it intuitively, but I cannot prove it formally. Anyone can help? This would also give me a hint on a methodology and "way of thinking" to solve the next ones.

So, the problem is the following. Let V be the space of functions v continuous on [0,1], with v' piecewise continuous and bounded on [0,1], and v(0) = v(1) = 0.
Demonstrate that, if a function w is continuous on [0,1], and the definite integral between 0 and 1 of w*v is zero for each v belonging to V, then w = 0 on all [0,1].

A proof by contradiction. Suppose that over an interval , and without loss of generality let .
Construct (as an exercise) a smooth function such that at the middle third of the interval , ascends to zero on the other two intervals, and outside of . Then we have